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S01.3 · Center, spread and shape

A free, illustrated module of 4 lessons and one interactive in S01 Exploratory Data Analysis: The mean; The median, quartiles and the five-number summary; Variance and standard deviation; Shape: z-scores, skew, modes and tails.

S01.3 L1 · Core 4 lessons · about 100 min Live · in review

Part of S01 Exploratory Data Analysis in the Statistics and Data Analysis track.

Three questions about one column of numbers: where is the middle, how far do the values stray from it, and what does the pile of values look like? Worked on the nine rents of the flats table, these four lessons build the mean as a balance point and the median as the middle of a sorted list, measure spread with the quartiles and with the standard deviation (and explain its divisor n - 1), then read shape from z-scores, skewness, modes and tails, so that you know which summary to trust when one penthouse pays three times the rest.

What you need first. Sum notation (M01.7), the flats table and its kinds of columns (S01.1), and population, sample, bias and the weighted mean of strata (S01.2). A little NumPy and pandas (P02) for the code.

Lessons

1. The mean S01.3.1 25 min

   The mean is the balance point of a dot plot. See why the deviations from it cancel, prove that it is the constant with the smallest squared error, rebuild it from group means with a weighted mean, and watch one penthouse drag it.

2. The median, quartiles and the five-number summary S01.3.2 25 min

   Sort the values and count to the middle: that is the median, and one wild value cannot drag it. Add the quartiles by a stated rule, then the range, the interquartile range and the five-number summary, and see why textbooks disagree on quartiles.

3. Variance and standard deviation S01.3.3 25 min

   Square every deviation from the mean, add the squares, divide by n - 1 and take the root: the standard deviation, back in the units of the data. See what shifts and changes of unit do to it, and why the divisor is n - 1.

4. Shape: z-scores, skew, modes and tails S01.3.4 25 min

   Two columns can share their mean and standard deviation and still look nothing alike. Standardise values into z-scores, measure lopsidedness with the skewness, read modes and tails from dot plots, and choose the summaries that fit the shape.

The interactive

S01.3-X1 · Balance and middle. Drag any of the nine rents along the number line. The mean, the band of one standard deviation and the quartile box redraw, and the read-out shows that the mean and the standard deviation follow a dragged value while the median and the interquartile range hold still. It lives in Variance and standard deviation.